Unconditionally secure bit commitment with flying qudits

In the task cryptographers call bit commitment, one party encrypts a prediction in a way that cannot be decrypted until they supply a key, but has only one valid key. Bit commitment has many applications, and has been much studied, but completely and provably secure schemes have remained elusive. Here we report a new development in physics-based cryptography which gives a completely new way of implementing bit commitment that is perfectly secure. The technique involves sending a quantum state (for instance one or more photons) at light speed in one of two or more directions, either along a secure channel or by quantum teleportation. Its security proof relies on the no-cloning theorem of quantum theory and the no-superluminal-signalling principle of special relativity.


II. INTRODUCTION
Alice and Bob participate in a stock market that trades at a specific physical site.Alice has a way of generating market predictions, which she wishes to demonstrate to Bob, in such a way that he can verify the accuracy of her predictions post hoc, but cannot possibly exploit them before the predicted events occur.Bob needs a guarantee that Alice's predictions were genuinely made at or before the point she claims they were, and were not retroactively altered post hoc.They can solve their dilemma with a suitably secure protocol for bit commitment.
Much attention has been devoted to the problem of bit commitment, which is a basic cryptographic task that is important per se, and has many applications to other more complex tasks.It also has intriguingly deep connections to fundamental physics, which have been uncovered in the search for bit commitment schemes whose security is guaranteed by the laws of physics alone (i.e.without any need for extra computational or technological assumptions).
Initially, work in this area focussed entirely on protocols based on non-relativistic quantum mechanics.Bennett and Brassard invented the first quantum bit commitment protocol [1], which (as they noted) is secure against both parties given current technology, but insecure if Alice has a quantum memory.Later attempts at unconditionally secure non-relativistic protocols (e.g.[2]) were ultimately shown to be futile by the celebrated results of Mayers [3][4][5] and of Lo and Chau [6,7], later further elaborated [8,9], which show that no unconditionally secure non-relativistic quantum bit commitment protocol exists.
This picture changes radically when we also exploit the signalling constraints implied by special relativity.The possibility of using relativistic signalling constraints for bit commitment was briefly discussed by Mayers [3], who suggested that his version of the no-go theorem should also apply to relativistic protocols.One strategy for bit commitment based on temporary relativistic signalling constraints was indeed shown to be insecure against quantum attacks [10].The discovery [11,12] that relativistic protocols can evade the Mayers and Lo-Chau no-go theorems thus came as a surprise.Encouragingly, such protocols are practical (although challenging) to implement with existing technology: ref. [12] describes a relativistic bit commitment protocol in which the parties can maintain bit commitments indefinitely, by exchanging classical information at a constant -and presently feasible -rate between two pairs of separated sites under their respective control.It is provably secure against all classical attacks and against Mayers and Lo-Chau's quantum attacks, and is conjectured to be unconditionally secure.However, there is as yet no complete security proof against general attacks.
Here we introduce a completely new technique for bit commitment, which relies essentially on the properties of quantum information in relativistic quantum theory -specifically, on the no-summoning theorem [13].It is provably unconditionally secure.In its simplest form it requires only one quantum state transmission per commitment.It illustrates a new way of exploiting cryptographically the control over physical information that special relativity and quantum theory together allow.This seems likely to find many other applications (e.g.[14]).
One interesting potential application of the present work is to high-frequency financial trading, in which light speed signalling constraints are already a significant and potentially disruptive factor [15].In a financial context, bit commitment is usually thought of as a technique for making encrypted predictions that can be decrypted and checked after the event.However, in an appropriately regulated framework, it could also be used to commit parties to trades without immediately making the trade public, or even without immediately informing one of the trading parties (who would have to have agreed to a regulatory framework in which the risks and disadvantages of such blind transactions were appropriately managed and compensated).The techniques described here thus illustrate, inter alia, that light speed signalling constraints also offer a tool for either traders or regulators to reshape market dynamics via secure commitments.In a possible future world in which conventional cryptosystems are rendered suspect by quantum computers or other developments, they may be the only reliably secure techniques.

III. BIT COMMITMENT WITH FLYING QUDITS
We begin by idealizing to simplify the presentation of the essential idea.We will relax these unphysical idealized assumptions later.We suppose that space-time is Minkowski and that nature is described by some appropriate relativistic version of quantum theory.We suppose that both parties, the committer (Alice) and the recipient (Bob), have arbitrarily efficient technology, limited only by physical principles.We suppose that all their operations and communications are error-free and that Alice can carry out quantum operations instantaneously.We also suppose they agree in advance on some space-time point P , to which they have independent secure access, where the commitment will commence.
We suppose they are independently and securely able to access every point1 in the causal future of P , and instantaneously process and exchange information there, and that each is able to keep information everywhere secure from the other unless and until they choose to disclose it. 2We suppose also that they can securely exchange quantum states at P , and at other relevant points to which they both have secure access.
We suppose too that Bob can keep a state private somewhere in the past of P and arrange to transfer it to Alice at P .Alice's operations on the state can then be kept private unless and until she chooses to return information to Bob at some point(s) in the future of P .We also suppose that Alice can send any relevant states at light speed in prescribed directions along secure quantum channels.
They also agree on a fixed inertial reference frame, and two opposite spatial directions within that frame.We initially simplify further by working in one space and one time dimension; we set c = 1 and take P to be the origin in the fixed frame coordinates (x, t) and the two spatial directions to be defined by the vectors v 0 = (−1, 0) and v 1 = (1, 0).
Before the commitment, Bob generates a random pure qudit state ρ ∈ C d , chosen from the uniform distribution, encoded in a physical system which (idealizing again) we take to be pointlike.He keeps it private until P , where he gives it to Alice.To commit to the bit i ∈ {0, 1}, Alice sends the state ρ along a secure channel at light speed in the direction v i , i.e. along the line L 0 = {(−t, t), t > 0} (for 0) or the line L 1 = {(t, t), t > 0} (for 1).
In the simplest implementation of this protocol, Alice's secure channel may be physically secured -for instance, a shielded region of free space.In this case, to fit the standard model for mistrustful cryptography, we consider the relevant channels as lying within Alice's secure laboratory.Alternatively, if Alice knows in advance the points at which she wishes to unveil her commitment, she can predistribute entangled states between P and these points, and implement a secure channel by teleporting the unknown state to a point on the relevant light ray, broadcasting the classical teleportation signal from P .Security here is guaranteed since the classical teleportation signals carry no information about either the transmitted state or (more importantly) the direction in which it is teleported.
For simplicity, we consider here the simplest implementation in which the state is directly securely transmitted.Alice can then unveil her commitment at any point along the transmitted light ray.To unveil a 0, Alice returns ρ to Bob at some point Q 0 on L 0 ; to unveil a 1, Alice returns ρ to Bob at some point Q 1 on L 1 .Bob then carries out the appropriate projective measurement to verify that the returned qudit is ρ; if he gets the correct answer, he accepts the commitment as honestly unveiled; if not (given that at this stage of the discussion we make the idealized assumption of no errors), he has detected Alice cheating.

A. Security against Bob
Given our assumptions, the protocol is evidently secure against Bob, who learns nothing about Alice's choice until the unveiling.

No perfect cheating strategy
Once Alice has carried out quantum operations of her choice at P , her strategy for optimizing the probability of successfully unveiling 0 is independent of the unveiling point Q 0 on L 0 (and similarly for 1).Although Alice is free to choose the points Q 0 and Q 1 , and may vary them depending on other relevant information that reaches her from the relevant past light cones, we may thus without loss of generality consider Q 0 and Q 1 as fixed.
Whatever operations and strategies she chooses, Alice cannot guarantee both that she will be able to unveil successfully at Q 0 if she (at Q 0 ) chooses to, and also that she will be able to unveil successfully at Q 1 if she (at Q 1 ) chooses.If she could, she would be able to guarantee a successful unveiling at both points, by following the appropriate strategies at both points.This would violate the no-cloning theorem, since she would be able to guarantee producing two copies of the unknown pure state ρ in the frame in which Q 0 and Q 1 are contemporaneous.More fundamentally, from an intrinsically relativistic perspective, she would be violating the no-summoning theorem [13], an intrinsic feature of relativistic quantum theory that extends the no-cloning theorem [16,17] and the no-signalling principle.To see this, note that she would be able to guarantee successfully responding to a summons from another party at Q 0 , requiring her to produce the state ρ at Q 0 , and also to a similar summons at Q 1 -and hence to both summonses, which is impossible.

Full security against cheating
Suppose Alice decides, at (or in the past of) P , that she wishes to retain as much freedom as possible in her choice of which bit to unveil, and is willing to accept that this may entail some risk of being caught cheating.Specifically, she wants to design a strategy which gives her a probability p 0 of successfully unveiling the bit 0 at Q 0 , should she (at Q 0 ) decide she wishes to, and a probability p 1 of successfully unveiling the bit 1 at Q 1 , should she (at Q 1 ) decide she wishes to.
Any quantum bit commitment protocol in which an honest committer can be certain of successful unveiling allows such strategies, in which the value of the unveiled bit is genuinely undetermined (not merely unknown to Alice) until the unveiling, for any probabilities p 0 and p 1 with p 0 + p 1 = 1.To achieve this, the committer simply has to prepare a state of the form where the | I state is input as the committed bit and the entangled | A state is stored until unveiling and then measured in the computational basis. 3Indeed, even classical bit commitment protocols can achieve a similiar outcome if the committer chooses their input randomly, without observing the random choice.In this case there is a definite committed bit value (and this is an important difference in some applications), but from Alice's perspective the unveiled bit remains unknown, with subjective probabilities obeying p 0 + p 1 = 1.
A sensible definition of cheating thus requires that a cheating Alice can at least ensure that p 0 + p 1 > 1.More precisely, we say a protocol with a security parameter N is unconditionally secure provided that, for any committing and unveiling strategies, we have p 0 + p 1 < 1 + (N ), where (N ) → 0 as N → ∞.
In our protocol, the security parameter is the qudit dimension d.Again, we can treat Q 0 and Q 1 as fixed.Suppose that Alice carries out some fixed operations at P , with a view to giving herself some chance of successful unveiling at either Q 0 or at Q 1 .Let ρ 0 be the state she generates at Q 0 if she subsequently follows the strategy that optimizes her chances of successfully unveiling there, and ρ 1 the corresponding state at Q 1 .Again, we can treat Alice's decisions whether to unveil at Q 0 and Q 1 as independent, and imagine that she chooses to unveil at both points.She then produces, at space-like separated points, the states ρ 0 and ρ 1 (which may and generally will be mixed), which, roughly speaking, are intended to be near-copies of the initial pure state ρ that are as faithful as possible.More precisely, since Bob tests the returned states ρ i by measuring the projector P ρ onto the pure state ρ, the probability p i of each state ρ i passing Bob's test is Tr(ρρ i ).We thus have p 0 + p 1 = Tr(ρρ 0 ) + Tr(ρρ 1 ) .
Effectively, in a reference frame in which Q 0 and Q 1 have the same time coordinate, Alice is attempting to implement 1 → 2 approximate cloning of the unknown state ρ.The expression on the right hand side, Tr(ρρ 0 ) + Tr(ρρ 1 ), is a standard measure of fidelity for this task.If Alice's process ensures that Tr(ρρ 0 ) = Tr(ρρ 1 ), she is implementing symmetric 1 → 2 cloning; otherwise, her scheme is asymmetric.Optimality bounds on the fidelity have been proved [20][21][22][23][24][25][26][27][28] in both cases.The symmetric bound suffices, since any asymmetric scheme can be symmetrized by randomization, without altering p 0 + p 1 (or in the 1 → N case, without altering N −1 i=0 p i ) , so there must be a symmetric scheme that is optimal by this measure.However, in the 1 → 2 case it seems a little more satisfying to use the explicit form for the asymmetric bounds.For 1 → 2 qudit cloning they give where Optimising via a Lagrange multiplier we find .
We thus indeed have unconditional security: the expression (p 0 + p 1 − 1) (Alice's "wiggle room") is bounded by a term that is O(d −1 ), i.e.O(exp(−Cn)), where n is the number of qubits that Alice and Bob exchange.

C. The non-ideal case
Realistically, the qudits used will not be pointlike physical systems, it will not be possible for Alice and Bob to have secure access to precisely the same space-time point to exchange a qudit, and Alice may not be able to transmit qudits at precisely speed c. 4 Realistically, too, Alice's and Bob's operations will take nonzero time.All of these limitations effectively introduce time delays at various stages of the protocol.
These delays can all be allowed for in a realistic definition of the task5 , without making the protocols insecure.However, they do affect the interval over which the commitment is guaranteed to be secure.To see this, suppose that Alice and Bob attempt to implement a protocol which ideally prescribes that the qudit should be returned at one of the points Q 0 , . . ., Q n−1 .Let Q 0 , . . ., Q n−1 be the points at which Bob would actually be able to complete his tests, given a real (non-ideal) implementation of the protocol, which allows Alice to return the qudit in specified finite regions of space-time to the future of the points Q i .Bob can then only be confident that Alice was committed at points lying outside the past light-cones of the Q i (i.e. that she has no guaranteed successful strategy for changing her commitment that relies on operations at any such points).In other words, although Alice may need in fact to be committed at the point P (and hence all points in its causal future) in order to implement her side of the protocol, the security analysis will only persuade Bob that she was committed by some point (or in some region) that lies in the future of P . 6f the Q i are suitably close to the Q i , this need not make any substantial difference, and thus may not be particularly relevant unless great precision over the place and time at which Alice became committed is required.However, when -as may be the case in our stock market example -a relatively precise timing of Alice's commitment is potentially crucial, it will be desirable to try to minimize the time delays arising in practical implementations.

D. Channel losses and errors
In reality, Alice and Bob's operations will not be error-free.With some nonzero probability, the qudit state will be altered somewhere between Bob's sending it to Alice and Alice returning it to Bob.Also, with some nonzero probability, the qudit will either be lost somewhere between Bob's transmitting it at some point P in the (presumably) near past of P and his receiving it at Q 0 or Q 1 (because Bob's or Alice's channels are not perfect) or will fail to be detected when received at Q 0 or Q 1 (because Bob's detectors are not perfect).
The possibility of losses can be countered by running N > 1 copies of the protocol, timed so that they are effectively run in parallel.Instead of supplying a single random qudit to Alice at P , Bob supplies a labelled sequence of N independently random qudits ρ 1 , . . ., ρ N , within a time interval short compared to the times and distances separating P and Q i (in lab frame).To commit the bit value i, Alice is required to send all the qudits to the point Q i , and return them from there, with their original labels, to Bob at Q i .Bob accepts the commitment as valid so long as his tests identify M qudits as those he originally sent, where M is statistically significantly above N/2, in the sense that the probabilities p 0 and p 1 for Alice being able to satisfy this test at Q 0 and Q 1 satisfy p 0 + p 1 ≤ 1 + (d, N, M ), where (d, N, M ) is suitably small (and can be made arbitrarily small by suitable parameter choices).
This strategy of redundant parallel commitment also gives a way of countering channel and detector errors.Standard (and more efficient) error correction techniques can also be used for this purpose.
The possibility of losses and errors is always an issue in quantum cryptography, and always adds to the practical challenges in implementing protocols.The protocols considered here are, of course, no exceptions.However, errors and losses raise no qualitatively new problems of principle for the protocols considered here.It is not the case, for example, that any nonzero error or loss rate makes it impossible to implement a secure version of the protocol.On the contrary, provided the loss and error rates are not too large, they can be countered by standard error correction techniques, reducing the probability of false positives (Alice successfully cheating) and false negatives (Bob being unable to accept Alice's honest unveiling) to arbitrarily small agreed levels -which is the best scenario possible in practical cryptography.

E. Relation to standard mistrustful cryptography
Note that relaxing our idealized assumptions allows us to fit the task definition into a standard cryptographic model [12] for mistrustful parties in Minkowski space-time.The non-idealized case allows us to drop the assumption that Alice and Bob each have independent secure access to every space-time point.Instead, we can adopt the standard assumption that Alice and Bob control suitably configured disjoint regions of space-time, their "laboratories".Each trusts the security of their laboratory and all devices contained within it, but need not trust anything outside their laboratory.
For example, we can take Alice's laboratory to be a connected region of space-time that includes, near its boundary, P and all allowed unveiling points Q i , and includes light ray segments corresponding to the secure channels joining P to each Q i : this allows Alice to receive a state at P and transmit it securely to any Q i .We can take Bob's laboratory to be a disjoint connected region of space-time that includes a point P in the near causal past of P , from which he sends the unknown state to P , and points Q i in the near causal future of each possible unveiling point Q i , to which Alice is supposed to send the unveiled state if she unveils at Q i .This allows Bob to generate the unknown state securely, transmit it to P , and then Alice to transmit it securely to some Q i of her choice, and return it to Bob at Q i , where he can test it securely.(See Figure 1.) As is standard in mistrustful cryptographic scenarios, we assume that Alice and Bob are the only relevant parties -no one else is trying to interfere with their communications -and that they have classical and quantum channels (which in principle can be made arbitrarily close to error-free) allowing them to send classical and quantum signals between the relevant points.
It is perhaps worth stressing here that, as with all such models, the main purpose is to show that the task is welldefined and implementable according to standard definitions.The model is not meant to prescribe how the protocol must be implemented in practice: for example, if Alice uses teleportation to transmit the qudit, her labs need not be connected.The point here is only to note one sensible way of implementing the protocol that uses only standard cryptographic assumptions.

F. Three dimensions and Committing more Data
We now consider three space dimensions, and suppose that Alice wants to commit to a value in {0, . . ., m − 1}, for some integer m ≥ 2. To achieve this we generalize the above protocol.We fix m distinct lightlike directions from the FIG.1: One implementation of the non-ideal protocol in 1 + 1 dimensions.(Not to scale.)Alice and Bob control disjoint regions of space-time, representing their respective secure laboratories.Bob sends the unknown state to Alice at point P , and she propagates it at near light speed to one of the points Qi, from where she sends it for Bob to receive at Q i .The arrows within Alice's lab denote light rays; the separations from P to the Qi are timelike to allow for delays at either end as well as possible subluminal communication.Note that, while Bob's laboratories here are disconnected, in more than one spatial dimension they can be connected without overlapping with Alice's.
commitment point P , defined by m distinct vectors v i in the agreed fixed inertial frame. 7To commit to the value i, Alice sends the received qudit from point P at light speed in the direction v i .To unveil her commitment, she returns the qudit to Bob at some point Q i on the relevant forward light ray from P .By the reverse triangle inequality for future-oriented timelike and lightlike vectors in Minkowski space, any such point Q i is spacelike separated from any other such Q j (for j = i), and so Alice's decisions about unveiling at Q i cannot be influenced by her communicating to Q i information she learns at any point Q j on any of the other light rays.
Suppose that Alice carries out some fixed operations at P , with a view to giving herself some chance of successful unveiling at each of the points Q i .Again, without loss of generality, we can analyse security while treating the Q i as fixed and (since an unveiling at Q i can be postponed to any future point on the relevant light ray) assuming they have the same time coordinate in some inertial frame F .Let ρ i be the state Alice generates at Q i if she subsequently follows the strategy that optimizes her chances of successfully unveiling there.Again, we can treat Alice's decisions whether to unveil at Q i as independent of her actions at Q j for j = i, and imagine that she chooses to unveil at all m points.In the frame F , her actions then become an attempt at 1 → m cloning.
The optimality bound [20][21][22][23] for symmetric 1 → m qudit cloning implies, via the symmetrization argument mentioned earlier, that We thus have unconditional security in this case also: the expression ( i p i − 1) is again bounded by a term that (for fixed m) is O(d −1 ), i.e. exponentially small in the number of qubits used.
In principle, this strategy of direction-dependent commitment allows an arbitrary amount of data to be securely committed and unveiled, using a single qudit, and within a fixed finite space-time volume.However, there are two important caveats here.
First, security requires that 2md −1 is small, meaning that the dimension d needs to scale at least8 linearly in m.Second, Alice needs to be able to transmit from P in m distinct directions, and Bob needs to be able to distinguish m distinct unveiling points on the relevant light rays.Alice thus needs to be able to specify transmission directions to within a solid angle9 small compared to m −1 , and Bob needs to be able to distinguish separations small compared to the distances between the corresponding unveiling points.These precisions are thus exponential in the number of bits committed, log 2 m.In realistic implementations, attaining such precisions should be considered as consuming resources that also scale at least exponentially in log 2 m for large m.

IV. DISCUSSION
The new bit commitment protocols described here are theoretically interesting in that they make use of a property of relativistic quantum theory -which seems most fundamentally characterised by the no-summoning principle [13] -that has not previously been exploited cryptographically, and which looks likely to find many other applications.Working out in full generality precisely which applications of bit commitment can be implemented using sequences of these protocols poses some interesting theoretical and practical challenges.
To implement the protocol efficiently and reliably requires transmitting qudits efficiently and reliably (by teleportation, or within cryptographically secure regions of space), at near light speed along pre-determined alternative paths.If one takes an optimistic view, common among quantum information scientists, technological barriers that currently prevent near-ideal implementation of quantum information processing and communication will ultimately be overcome.At some point, the transition to practical quantum information processing and communication will take place -and at that point the practicality of quantum computing motivates a transition from (newly vulnerable) classical cryptographic protocols to secure quantum protocols.This is sufficient motivation for the protocols described here: if and when practical and reliable quantum information processing technology emerges, they will be both practical and practically relevant.
That said, it is worth reviewing how practical the protocols already are with present technology.For a proof that the basic concept can be implemented, Alice needs only be able to route photons at near light speed along alternative paths -which may be through free space -in regions which we can assume lie within her laboratory (and thus are secure), and return the photons to Bob's detectors.Moreover, for a partial proof of concept, one might initially accept an unreliable implementation, in which Alice's commitments are sometimes verified by Bob but sometimes (because of losses or errors) unverified.One might too accept implementations in which the photons travel at significantly lower than light speed (along optical fibres of high refractive index): such implementations still guarantee a finite (though shorter) duration commitment, so long as the allowed unveiling points are space-like separated.Under some or all of these relaxed assumptions, the protocol seems well within the scope of current technology.How reliably it can presently be implemented, over what ranges, with what levels of security, and how small the transmission and other delays can be made compared to the ideal protocol, are open questions, which we offer as challenges to the ingenuity of experimentalist colleagues.
Returning to the optimistic long term view of quantum information technology, it seems to us that a plausible future cryptographic environment will require unconditional security for bit commitments of short duration, and that these protocols may turn out to be the most efficient and easiest solutions.For example, one can imagine short term commitments relating to a stock market being made and unveiled by two parties exchanging photon signals that are transmitted along independently secure links.For such applications, it will be particularly interesting to explore and analyse the security and efficiency of chaining protocols and redundant encoding (discussed in the Appendix).
Previous protocols [11,12] that make use of the Minkowski causal structure to implement secure bit commitment rely on the fact that data introduced at one site is not available to an agent at a space-like separated site.As noted earlier, these protocols are immune to Mayers' and Lo-Chau's cheating strategies.The MLC strategies do, in principle, define an operation that the committer could carry out in order to cheat the protocols of Refs.[11,12] by altering their commitment.However, this operation always depends on data introduced at a space-like separated site, and so is never knowable by the committer.
The protocols introduced here highlight another significant limitation of Mayers' and Lo-Chau's no-go theorems that appears to have previously gone unnoticed.Namely, in relativistic quantum theory, the unitary operation required for a MLC attack can be known to both parties but impossible to implement physically, as it represents a spacelike translation that would violate causality.This reinforces the point -if any reinforcement were needed -that these celebrated and fundamental theorems are correctly understood as applying specifically to protocols that rely only on non-relativistic quantum mechanics.They can be extended to some important classes of relativistic protocolsfor example those in which the unveiling point takes place at a fixed point in the causal future of all the operations carried out in the commitment phase -but do not apply to general protocols based on relativistic quantum theory.
Like all intrinsically quantum bit commitment protocols [18,19], the protocols considered here allow the committer to commit a bit in quantum superposition, which is "collapsed" to a classical bit value only when unveiled.For some important applications of bit commitment, this feature makes no essential difference.For example, if the bit commitment encodes a prediction, allowing the committer to make a superposition of predictions merely gives them the freedom to add a random element to their prediction -which is also possible with any classical bit commitment scheme, since the committer can always randomize their input.Indeed, for some intrinsically quantum applications, in which the ultimate aim is to unveil a quantum state, it may be a positive advantage that the scheme allows commitment of a qubit rather than a bit.On the other hand, our scheme cannot be securely used in applications of bit commitment in which it is crucial that the committed bit takes a classical value fixed at (or before) commitment.It bears reemphasizing that this is a quite general feature of intrinsically quantum bit commitment schemes, and quite separate from the cheating possibilities pointed out by Mayers and Lo-Chau. 10 The protocols we have described are unconditionally secure, but not necessarily optimally efficient, in the sense that they achieve optimal security for given resources.It will be interesting to investigate the range of possible strategies and their efficiencies.One particularly interesting possibility is to consider protocols based on summoning an entangled state.For example, Bob could generate a randomly chosen maximally entangled pair of qudits, and initiate the commitment protocol by giving one of them to Alice -who must later return it as above -while he keeps the other.To verify the unveiled commitment in this case, he needs to recombine his stored qudit with the returned qudit at a single site using secure quantum channels, or else carry out a non-local measurement.Either way, he cannot generally verify with certainty at the point of unveiling, but can do so at a future point. 11 These protocols have the advantage of simplicity: they require only acting on a qudit with a simple quantum operation to determine its transmission direction, and then transmitting it.The fact that the unveiling location needs to be on a light ray from the commitment point, in a direction depending on the committed bit, is, however, potentially a disadvantage.Whether it is problematic, and if so how much so, very much depends on the scenario in which the bit commitment is being used.
For example, if the trading centre of a stock market is physically localized around the point P , and Alice wants to commit at time 0 predictions of prices at a pre-agreed time t > 0, which she is happy to unveil as soon as the data cannot be exploited by Bob, the protocol may be perfectly adequate, so long as they can securely exchange information at suitable points distance (t/2) from P in stock market rest frame (i.e.so long as t is not too large).
On the other hand, if Alice makes some prediction at time 0 which she may or may not wish to unveil, depending on details of the market's behaviour after t = 0 -perhaps, for example, because the prediction is of some future events conditioned on others, and she wishes to give away no information about her predictive abilities unless the conditioning events take place -then these protocols alone do not suffice.Alice's decision to unveil on any light ray from P must necessarily be made in ignorance of events at the location of P at later times.

B. Chaining commitments
Some flexibility in the location of the unveiling point, relative to the commitment point, can be achieved by chaining a number of our commitment protocols in sequence.For example, consider the following extended protocol.Alice and Bob fix P , as above, and a time interval T in their fixed inertial frame; they also fix the qudit dimension d.At P , Bob initiates the protocol by giving Alice an unknown qudit, and she commits a bit by sending it along the appropriate light ray to points Q 0 or Q 1 , which have time coordinate T .Suppose, for example, she commits to 0. At Q 0 , she returns the qudit to Bob, after acting on it with a randomly chosen d-dimensional teleportation operation.This operation is specified by an integer j in {0, . . ., d 2 − 1}.Bob initiates a new data commitment protocol at Q 0 , giving an Alice a qudit of pre-agreed dimension d (here d > d), which is used to commit to the value j.At Q 1 , Alice and Bob go through the same operations, but here Alice simply returns a randomly chosen dummy qudit to Bob, and commits to a randomly chosen value j .This procedure can be iterated any number of times.
To unveil, Alice returns the qudits from the final commitment without any randomization operation, allowing Bob to infer the operations necessary to derandomize the qudits returned earlier, and hence the originally committed bit.This procedure has the advantage that Alice's final unveiling location (although randomly determined) will generally be timelike rather than lightlike separated from P .By varying the parameters of the chained protocol the distribution of separations can be optimized (within the space of possible distributions) for any given task.Of course, chaining the protocol requires extra quantum operations and communications from both parties, which moreover grow exponentially in the number of iterations.It may perhaps be possible to mitigate this by adapting techniques used [12] to eliminate exponential blow-ups for classical relativistic bit commitment protocols [11], which allow one to truncate successive linked commitments after a fixed number of iterations and replace them by newly started commitments, while retaining security.
This strategy remains to be fully explored.We mention chained protocols here to note that the strategy of making bit commitments by appropriate light speed transmission -or indeed using similar techniques for other cryptographic applications -is not as inflexible as it may initially seem from considering the simplest (unchained) protocols.We leave the analyses of resource optimization, practicality and security for chained protocols for a future discussion.

C. Redundant encoding
Another valuable strategy is to run two or more implementations of the protocol in parallel, using different bit codings.For example, in the basic one-dimensional protocol, Bob can give Alice two independently randomly generated unknown pure states, ρ 0 and ρ 1 , at the same point P .She follows the protocol described above with ρ 0 , sending it along the line L i to commit to bit value i; she follows the protocol with reversed conventions for ρ 1 , sending it along the line L i+1 (where i + 1 is defined modulo 2) to commit to bit value i.This allows her to unveil the commitment at any point Q 0 on L 0 or any point Q 1 on L 1 , regardless of the committed bit value.To prevent her cheating, she must also be required to unveil at some point on the opposite line.
Giving Alice this freedom ensures that the bit can be unveiled on an agreed light ray -or indeed, if the parties wish, at a pre-agreed point on that light ray.Note that if Bob accepts such an unveiling immediately Alice has scope for a form of temporary cheating. 12For example, she could cheat by sending both ρ 0 and ρ 1 along the agreed light ray L 0 and then decide which to return to Bob at the unveiling point Q 0 .However, any such cheating will not allow her to produce a consistent unveiling at Q 1 .Her deception will thus eventually become evident to Bob. 13In many scenarios -for example when individual commitments are part of an ongoing series of transactions of value to both parties -the opportunity to deceive Bob temporarily is of little value to Alice.Of course, in scenarios where Bob can suffer significant loss through temporary deception, and cannot recover it once the cheating is exposed, he should not accept Alice's unveiling until (i.e. at spacetime points where) he can combine all the relevant unveiling data.
Note that -with the above caveats about temporary cheating -redundant encoding can be combined with the chaining strategy described above, allowing Alice and Bob to ensure that the original commitment is unveiled at any point they wish in the causal future of P .In our stock market example, for instance, they can ensure that the commitment is unveiled at the market location at any agreed future time.